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Oscillatory critical amplitudes in hierarchical models and the tail of the Harris random variable

Oscillatory critical amplitudes have been repeatedly observed in hierarchical models and, in the cases that have been taken into consideration, these oscillations are so small to be hardly detectable. Hierarchical models are tightly related to iteration of maps and, in fact, very similar phenomena have been repeatedly reported in many fields of mathematics, like combinatorial evaluations and discrete branching processes. It is precisely in the context of branching processes with bounded off-spring that T. Harris, in 1948, first set forth the possibility that the logarithm of the moment generating function of the rescaled population size, in the super-critical regime, does not grow near infinity as a power, but it has an oscillatory prefactor. These oscillations have been observed numerically only much later and, while the origin is clearly tied to the discrete character of the iteration, the amplitude size is not so well understood. The purpose of this note is to reconsider the issue for hierarchical models and in what is arguably the most elementary setting -- the pinning model -- that actually just boils down to iteration of polynomial maps (and, notably, quadratic maps). In this note we show that the oscillatory critical amplitude for pinning models and the oscillating pre factor connected to the Harris random variable coincide. Moreover we make explicit the link between these oscillatory functions and the geometry of the Julia set of the map, making thus rigorous and quantitative some ideas set forth by B. Derrida, C. Itzykson and J. M. Luck (CMP 1984).